Q.Cut Down Pencil Cactus.. Small Dead Twigs On Ground Are They Toxic To Puppy
Cannot get all small dead Twigs up. Are they toxic when dead and brittle

Certified GKH Gardening Expert
It's the sap or latex that is irritating to the mouth and stomach and can cause vomiting. After extensive research, I couldn't find elaboration on whether the irritant exists after the plant is dead and brittle, but I would suspect the danger is greatly reduced. (However, I did read that you can still get poison ivy when the plant is dead.)
I would err on the side of caution and put soil, pebbles, or mulch on the area where the twigs fell and steer the puppy clear of the area.
https://www.aspca.org/pet-care/animal-poison-control/toxic-and-non-toxic-plants/pencil-cactus